Frequency of Arrival (FOA) measurements provide bearing angles that can be used for localization provided the known position, velocity and frequency of the moving emitter platform. In the well-known Doppler equation, the frequency observed by a receiving platform is shifted in a nonlinear fashion relative to the emitter's carrier frequency fe and the geometric relationship between the receiver and the emitting RF platform, is shown below in Equation 1:
                              Δ          ⁢                                          ⁢          f                =                              (                                          v                ⁢                                                                  ⁢                cos                ⁢                                                                  ⁢                θ                            c                        )                    ⁢                      f            e                                              Equation        ⁢                                  ⁢        1            where Δf the Doppler shift, v is the speed of the receiver relative to the emitter platform, θ is the cone angle between the receiver and the platform velocity vector, and c is the speed of light. Multiple Doppler observations can be triangulated and used to solve for the unknown receiver location.
To date, the most common methods for geolocation employ multilateration (i.e., multiple emitters). Typically these methods employ measurements of Time Difference of Arrival (“TDOA”) and/or Frequency Difference of Arrival (“FDOA”) of RF signals received from at least two emitters. For example, the 3D position of a receiver may be determined by measuring the TDOA of a signal from four or more spatially separate emitters located different distances from the receiver. If the emitting platform and receiver are in relative motion with respect to one another, FDOA may be employed either independently or in addition to TDOA to determine the receiver position based on \ the observed Doppler information. The popular global positioning system (GPS) uses the TDOA measurements collected from four or more satellites to act as RF beacons for a GPS receiver.
FDOA uses the difference of frequencies collected simultaneously from at least two different moving emitters. As shown in Equation 2, below, the FDOA observable Δω describes the Doppler shift caused by the relative change in range between the platform and two emitters:
                              Δ          ⁢                                          ⁢          ω                =                                            f              e                        c                    ⁢                      ⅆ                          ⅆ              t                                ⁢                      (                                          r                1                            -                              r                2                                      )                                              Equation        ⁢                                  ⁢        2            where the variable rk represents the range between the receiver k and an emitter.
A drawback of FDOA-based receiver location algorithms is that they require at least two or more simultaneous emitters. Furthermore, local oscillators (sensors) used in receivers such as cellular phones or handheld GPS devices often impart a bias and drift into their frequency measurements. For example, a given sensor may have an RF measurement bias, meaning an offset between the reported measurement and the actual frequency sampled. Further, this offset may have a stationary component, and also a component that drifts or changes over time. For example, as the temperature of the sensor changes, the offset may increase or decrease. This can cause a large localization error if not corrected.
This correction is sometimes done using known calibration tones present in the collected data, however, this requires even more resources, including a potentially cooperative emitter, and further motivates a robust, single beacon, single receiver FOA localization solution.
The FOA measurement equation is derived directly from Equation 1 by rearranging terms to isolate the observed Doppler-shifted frequency fobs from the unobserved Doppler shift Δf that depends on the emitter's carrier frequency fe. Furthermore, the cosine term can be expanded to explicitly contain the state variables of receiver location [xr,yr]T and emitter frequency fe by writing out the dot product between the emitter velocity [vx,vy]T and the line of sight between the receiver and the emitting platform [x,y]T:
                              f          obs                =                                            [                              1                -                                                      1                    c                                    ⁢                                                                                                              (                                                      x                            -                                                          x                              r                                                                                )                                                ⁢                                                  v                          x                                                                    +                                                                        (                                                      y                            -                                                          y                              r                                                                                )                                                ⁢                                                  v                          y                                                                                                            c                      ⁢                                                                                                                                  (                                                              x                                -                                                                  x                                  r                                                                                            )                                                        2                                                    +                                                                                    (                                                              y                                -                                                                  y                                  r                                                                                            )                                                        2                                                                                                                                                          ]                        ⁢                          f              e                                -                      f            r                                              Equation        ⁢                                  ⁢        3            
A primary reason that the FOA method is generally not considered an operational localization option is due to its tendency to produce unreliable results.
One reason for this is the fact that many commercial receiver systems do not have strict tolerances on oscillator stability which means that a receiver's frequency estimate can be both biased and drift significantly during normal operation. Failing to account for frequency drift observed over the course of triangulation time required to determine a unique solution will cause gross inaccuracies in location estimates. However, including a non-stationary drift term in the location solution eliminates the suitability of Gauss Newton and similar techniques, because it requires a stationary state model. This motivates the need for a new solution that can efficiently represent and track in a non-linear, non-convex, multimodal state space.